Solve the following:
Q: A company holds spare parts for its car maintenance service. There is a steady demand for these parts. If the company orders large numbers once a year, then they have to pay considerable warehouse costs to stock them. If they order small numbers very frequently then they have to pay considerable admin costs for processing all the orders. Mathematical analysis shows that the total cost (in pounds), C, of ordering and stocking the spare parts is given by:
C = 5000/q + 20q
where q is the number of parts ordered with each order.
(i) Find dC/dq
(ii) Find the turning point at C (i.e. what value of q makes dC/dq equal to 0).
(iii) By evaluating d2C/dq2 at the turning point, show that the turning point is a minimum.
(iv) Calculate the minimum cost that the company will have to pay for ordering and stocking.